Transmission of harmonic functions through quasicircles on compact Riemann surfaces

Let $R$ be a compact surface and let $Γ$ be a Jordan curve which separates $R$ into two connected components $Σ_1$ and $Σ_2$. A harmonic function $h_1$ on $Σ_1$ of bounded Dirichlet norm has boundary values $H$ in a certain conformally invariant non-tangential sense on $Γ$. We show that if $Γ$ is a quasicircle, then there is a unique harmonic function $h_2$ of bounded Dirichlet norm on $Σ_2$ whose boundary values agree with those of $h_1$. Furthermore, the resulting map from the Dirichlet space of $Σ_1$ into $Σ_2$ is bounded with respect to the Dirichlet semi-norm.